An elementary proof for the dimension of the graph of the classical Weierstrass function
Abstract
Let $W_{\lambda,b}(x)=\sum_{n=0}^\infty\lambda^n g(b^n x)$ where $b\geqslant2$ is an integer and $g(u)=\cos(2\pi u)$ (classical Weierstrass function). Building on work by Ledrappier (1992), Baránsky, Bárány and Romanowska (2013) and Tsujii (2001), we provide an elementary proof that the Hausdorff dimension of $W_{\lambda,b}$ equals $2+\frac{\log\lambda}{\log b}$ for all $\lambda\in(\lambda_b,1)$ with a suitable $\lambda_b<1$. This reproduces results by Baránsky, Bárány and Romanowska without using the dimension theory for hyperbolic measures of Ledrappier and Young (1985,1988), which is replaced by a simple telescoping argument together with a recursive multiscale estimate.
 Publication:

arXiv eprints
 Pub Date:
 June 2014
 arXiv:
 arXiv:1406.3571
 Bibcode:
 2014arXiv1406.3571K
 Keywords:

 Mathematics  Dynamical Systems;
 37D20;
 37D45;
 37G35;
 37H20
 EPrint:
 The proof of Proposition 3.3 is clarified and a mistake is corrected